The codegree threshold of K4−K_4^-

The codegree threshold $\mathrm{ex}_2(n, F)$ of a $3$-graph $F$ is the minimum $d=d(n)$ such that every $3$-graph on $n$ vertices in which every pair of vertices is contained in at least $d+1$ edges contains a copy of $F$ as a subgraph. We study $\mathrm{ex}_2(n, F)$ when $F=K_4^-$, the $3$-graph on $4$ vertices with $3$ edges. Using flag algebra techniques, we prove that if $n$ is sufficiently large then $\mathrm{ex}_2(n, K_4^-)\leq (n+1)/4$. This settles in the affirmative a conjecture of Nagle from 1999. In addition, we obtain a stability result: for every near-extremal configuration $G$, there is a quasirandom tournament $T$ on the same vertex set such that $G$ is close in the edit distance to the $3$-graph $C(T)$ whose edges are the cyclically oriented triangles from $T$. For infinitely many values of $n$, we are further able to determine $\mathrm{ex}_2(n, K_4^-)$ exactly and to show that tournament-based constructions $C(T)$ are extremal for those values of $n$.Comment: 31 pages, 7 figures. Ancillary files to the submission contain the information needed to verify the flag algebra computation in Lemma 2.8. Expands on the 2017 conference paper of the same name by the same authors (Electronic Notes in Discrete Mathematics, Volume 61, pages 407-413

Limits of Order Types

The notion of limits of dense graphs was invented, among other reasons, to attack problems in extremal graph theory. It is straightforward to define limits of order types in analogy with limits of graphs, and this paper examines how to adapt to this setting two approaches developed to study limits of dense graphs. We first consider flag algebras, which were used to open various questions on graphs to mechanical solving via semidefinite programming. We define flag algebras of order types, and use them to obtain, via the semidefinite method, new lower bounds on the density of 5- or 6-tuples in convex position in arbitrary point sets, as well as some inequalities expressing the difficulty of sampling order types uniformly. We next consider graphons, a representation of limits of dense graphs that enable their study by continuous probabilistic or analytic methods. We investigate how planar measures fare as a candidate analogue of graphons for limits of order types. We show that the map sending a measure to its associated limit is continuous and, if restricted to uniform measures on compact convex sets, a homeomorphism. We prove, however, that this map is not surjective. Finally, we examine a limit of order types similar to classical constructions in combinatorial geometry (Erdos-Szekeres, Horton...) and show that it cannot be represented by any somewhere regular measure; we analyze this example via an analogue of Sylvester\u27s problem on the probability that k random points are in convex position

Rainbow triangles in three-colored graphs

Erdős and Sós proposed the problem of determining the maximum number F(n) of rainbow triangles in 3-edge-colored complete graphs on n vertices. They conjectured that F(n) = F(a) + F(b) + F(c) + F(d) + abc + abd + acd + bcd, where a + b + c + d = n and a, b, c, d are as equal as possible. We prove that the conjectured recurrence holds for sufficiently large n. We also prove the conjecture for n = 4k for all k ≥ 0. These results imply that lim F(n)/((n)(3)) = 0.4, and determine the unique limit object. In the proof we use flag algebras combined with stability arguments